The Internet is an interconnection of computer networks around the world. This network allows a huge number of public, private, academic, government, and business networks to communicate and exchange data on a local to global scale.
Wi-Fi is a technology that allows wireless electronic devices (smartphones, tablets, laptops etc) to connect to a local access point, (ie the router) over a certain range. It is also one of a few methods of accessing the INTERNET.
The INTERNET is a network of millions of computers communicating to each other, it is the language by which computers talk to each other. Wi-Fi is merely a means to access that vast network.
NB: Having a fast internet connection doesn’t guarantee an all-around good internet experience. This is because the internet is a community, and the interaction between any two parties depends on both. If you download a file from a party with a slow connection to the internet, the downloading process still takes a long time and unfortunately, there’s nothing you can do about it.
MODEM vs. ROUTER
Your router creates a network between the computers in your home.
The MODEM connects that network. When you are connecting to Wi-Fi, you are really connecting to your ROUTER, which forwards traffic between the INTERNET and your computer. Many internet providers (such as Bundu NetworX) offer a combined modem/router unit that performs both these functions in one device.